Explanation of a phil. of sci. prompt & suggestions for phil. of math books

The prompt is this: Chose an actual example of scientific explanation, and use it to evaluate Hempel's and Ruben's accounts of explanation.

I am not understanding the second part, how I am supposed to use an example to evaluate Hempel's and Ruben's accounts of explanation (Hempel's DN model and Ruben's causal single statement view of explanation). Am I supposed to present the example in the context of Hempel's and Ruben's accounts of explanation? Would appreciate some clarification.

Secondly, I was wonder if you guys knew of any introductory or somewhat moderate treatment of philosophy of math. I think I have a decent math background at the moment but not graduate school level. Are there any accessible and readable material of that field?

Secondly, I was wonder if you guys knew of any introductory or somewhat moderate treatment of philosophy of math. I think I have a decent math background at the moment but not graduate school level. Are there any accessible and readable material of that field?

You can start out with the really short book "Introduction to Mathematical Philosophy" by Bertrand Russell (available for free online here), and if you like that you can read his longer book "Principles of Mathematics", available for free online here (not to be confused with his three-volume symbolic treatise the Principia Mathematica). I also recommend "Philosophy of Mathematics: Selected Readings" by Paul Benacerraf and Hilary Putnam (if you read this book, make sure you read the introduction, which is excellent), which has excerpts from the books of different philosophers of mathematics.

There are a variety of philosophies of mathematics, which have dramatically different views on what mathematics really is. By far the most popular view (among mathematicians and philosophers of mathematics at least) is known as Mathematical Platonism, according to which mathematics consists of statements that are objectively true and are independent of the human mind. To sum up, in this view mathematics is discovered, not invented. Gottlob Frege famously held this view, and he tried to argue that all mathematics can be reduced to logical tautologies. I recommend reading his short book "The Foundations of Arithmetic".

On the other extreme, there are the people who say that mathematics is just an arbitrary creation of the human mind. The people who believed this used to be fringe people known as constructivists or intuitionists, like Brouwer and Heyting, who tried to create a severely restricted form of mathematics which only allowed the parts of mathematics which could be directly justified by human mental activity. But nowadays there are people who try to explain all of mathematics, not just some restricted version of it, based on human psychology. You can read the book "Where Mathematics Comes From" by George Lakoff for a defense of that position.

I happen to be in the Platonist camp, but maybe you'll prefer something else. There are definitely a lot more views to choose from. For instance, there is physism, which says that humans invented mathematics not as some arbitrary creation of their minds, but by observing patterns in the physical world (like you put one rock next to another rock, and you find that there are now two rocks, so you conclude that 1+1=2). This view goes back to Aristotle, but Roland Omnes recently wrote a nice book "Converging Realities" concerning this.

The philosophy of mathematics has been fascinating to me for a long time, so feel free to ask any questions about it, or if you want more book recommendations tailored to your interests in the philosophy of math.